3a8617a83f
-- Begin comments from J.T. Conklin: The most significant improvement is the addition of "float" versions of the math functions that take float arguments, return floats, and do all operations in floating point. This doesn't help (performance) much on the i386, but they are still nice to have. The float versions were orginally done by Cygnus' Ian Taylor when fdlibm was integrated into the libm we support for embedded systems. I gave Ian a copy of my libm as a starting point since I had already fixed a lot of bugs & problems in Sun's original code. After he was done, I cleaned it up a bit and integrated the changes back into my libm. -- End comments Reviewed by: jkh Submitted by: jtc
634 lines
22 KiB
Groff
634 lines
22 KiB
Groff
.\" Copyright (c) 1985 Regents of the University of California.
|
|
.\" All rights reserved.
|
|
.\"
|
|
.\" Redistribution and use in source and binary forms, with or without
|
|
.\" modification, are permitted provided that the following conditions
|
|
.\" are met:
|
|
.\" 1. Redistributions of source code must retain the above copyright
|
|
.\" notice, this list of conditions and the following disclaimer.
|
|
.\" 2. Redistributions in binary form must reproduce the above copyright
|
|
.\" notice, this list of conditions and the following disclaimer in the
|
|
.\" documentation and/or other materials provided with the distribution.
|
|
.\" 3. All advertising materials mentioning features or use of this software
|
|
.\" must display the following acknowledgement:
|
|
.\" This product includes software developed by the University of
|
|
.\" California, Berkeley and its contributors.
|
|
.\" 4. Neither the name of the University nor the names of its contributors
|
|
.\" may be used to endorse or promote products derived from this software
|
|
.\" without specific prior written permission.
|
|
.\"
|
|
.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
|
|
.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
|
.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
|
.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
|
|
.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
|
.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
|
|
.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
|
|
.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
|
.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
|
|
.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
|
|
.\" SUCH DAMAGE.
|
|
.\"
|
|
.\" from: @(#)math.3 6.10 (Berkeley) 5/6/91
|
|
.\" $Id: math.3,v 1.4 1994/02/25 19:43:56 jtc Exp $
|
|
.\"
|
|
.TH MATH 3M "May 6, 1991"
|
|
.UC 4
|
|
.ds up \fIulp\fR
|
|
.ds nn \fINaN\fR
|
|
.de If
|
|
.if n \\
|
|
\\$1Infinity\\$2
|
|
.if t \\
|
|
\\$1\\(if\\$2
|
|
..
|
|
.SH NAME
|
|
math \- introduction to mathematical library functions
|
|
.SH DESCRIPTION
|
|
These functions constitute the C math library,
|
|
.I libm.
|
|
The link editor searches this library under the \*(lq\-lm\*(rq option.
|
|
Declarations for these functions may be obtained from the include file
|
|
.RI < math.h >.
|
|
The Fortran math library is described in ``man 3f intro''.
|
|
.SH "LIST OF FUNCTIONS"
|
|
.sp 2
|
|
.nf
|
|
.ta \w'copysign'u+2n +\w'infnan.3m'u+10n +\w'inverse trigonometric func'u
|
|
\fIName\fP \fIAppears on Page\fP \fIDescription\fP \fIError Bound (ULPs)\fP
|
|
.ta \w'copysign'u+4n +\w'infnan.3m'u+4n +\w'inverse trigonometric function'u+6nC
|
|
.sp 5p
|
|
acos sin.3m inverse trigonometric function 3
|
|
acosh asinh.3m inverse hyperbolic function 3
|
|
asin sin.3m inverse trigonometric function 3
|
|
asinh asinh.3m inverse hyperbolic function 3
|
|
atan sin.3m inverse trigonometric function 1
|
|
atanh asinh.3m inverse hyperbolic function 3
|
|
atan2 sin.3m inverse trigonometric function 2
|
|
cabs hypot.3m complex absolute value 1
|
|
cbrt sqrt.3m cube root 1
|
|
ceil floor.3m integer no less than 0
|
|
copysign ieee.3m copy sign bit 0
|
|
cos sin.3m trigonometric function 1
|
|
cosh sinh.3m hyperbolic function 3
|
|
erf erf.3m error function ???
|
|
erfc erf.3m complementary error function ???
|
|
exp exp.3m exponential 1
|
|
expm1 exp.3m exp(x)\-1 1
|
|
fabs floor.3m absolute value 0
|
|
floor floor.3m integer no greater than 0
|
|
hypot hypot.3m Euclidean distance 1
|
|
ilogb ieee.3m exponent extraction 0
|
|
infnan infnan.3m signals exceptions
|
|
j0 j0.3m bessel function ???
|
|
j1 j0.3m bessel function ???
|
|
jn j0.3m bessel function ???
|
|
lgamma lgamma.3m log gamma function; (formerly gamma.3m)
|
|
log exp.3m natural logarithm 1
|
|
log10 exp.3m logarithm to base 10 3
|
|
log1p exp.3m log(1+x) 1
|
|
pow exp.3m exponential x**y 60\-500
|
|
remainder ieee.3m remainder 0
|
|
rint floor.3m round to nearest integer 0
|
|
scalbn ieee.3m exponent adjustment 0
|
|
sin sin.3m trigonometric function 1
|
|
sinh sinh.3m hyperbolic function 3
|
|
sqrt sqrt.3m square root 1
|
|
tan sin.3m trigonometric function 3
|
|
tanh sinh.3m hyperbolic function 3
|
|
y0 j0.3m bessel function ???
|
|
y1 j0.3m bessel function ???
|
|
yn j0.3m bessel function ???
|
|
.ta
|
|
.fi
|
|
.SH NOTES
|
|
In 4.3 BSD, distributed from the University of California
|
|
in late 1985, most of the foregoing functions come in two
|
|
versions, one for the double\-precision "D" format in the
|
|
DEC VAX\-11 family of computers, another for double\-precision
|
|
arithmetic conforming to the IEEE Standard 754 for Binary
|
|
Floating\-Point Arithmetic. The two versions behave very
|
|
similarly, as should be expected from programs more accurate
|
|
and robust than was the norm when UNIX was born. For
|
|
instance, the programs are accurate to within the numbers
|
|
of \*(ups tabulated above; an \*(up is one \fIU\fRnit in the \fIL\fRast
|
|
\fIP\fRlace. And the programs have been cured of anomalies that
|
|
afflicted the older math library \fIlibm\fR in which incidents like
|
|
the following had been reported:
|
|
.RS
|
|
sqrt(\-1.0) = 0.0 and log(\-1.0) = \-1.7e38.
|
|
.br
|
|
cos(1.0e\-11) > cos(0.0) > 1.0.
|
|
.br
|
|
pow(x,1.0)
|
|
.if n \
|
|
!=
|
|
.if t \
|
|
\(!=
|
|
x when x = 2.0, 3.0, 4.0, ..., 9.0.
|
|
.br
|
|
pow(\-1.0,1.0e10) trapped on Integer Overflow.
|
|
.br
|
|
sqrt(1.0e30) and sqrt(1.0e\-30) were very slow.
|
|
.RE
|
|
However the two versions do differ in ways that have to be
|
|
explained, to which end the following notes are provided.
|
|
.PP
|
|
\fBDEC VAX\-11 D_floating\-point:\fR
|
|
.PP
|
|
This is the format for which the original math library \fIlibm\fR
|
|
was developed, and to which this manual is still principally
|
|
dedicated. It is \fIthe\fR double\-precision format for the PDP\-11
|
|
and the earlier VAX\-11 machines; VAX\-11s after 1983 were
|
|
provided with an optional "G" format closer to the IEEE
|
|
double\-precision format. The earlier DEC MicroVAXs have no
|
|
D format, only G double\-precision. (Why? Why not?)
|
|
.PP
|
|
Properties of D_floating\-point:
|
|
.RS
|
|
Wordsize: 64 bits, 8 bytes. Radix: Binary.
|
|
.br
|
|
Precision: 56
|
|
.if n \
|
|
sig.
|
|
.if t \
|
|
significant
|
|
bits, roughly like 17
|
|
.if n \
|
|
sig.
|
|
.if t \
|
|
significant
|
|
decimals.
|
|
.RS
|
|
If x and x' are consecutive positive D_floating\-point
|
|
numbers (they differ by 1 \*(up), then
|
|
.br
|
|
1.3e\-17 < 0.5**56 < (x'\-x)/x \(<= 0.5**55 < 2.8e\-17.
|
|
.RE
|
|
.nf
|
|
.ta \w'Range:'u+1n +\w'Underflow threshold'u+1n +\w'= 2.0**127'u+1n
|
|
Range: Overflow threshold = 2.0**127 = 1.7e38.
|
|
Underflow threshold = 0.5**128 = 2.9e\-39.
|
|
NOTE: THIS RANGE IS COMPARATIVELY NARROW.
|
|
.ta
|
|
.fi
|
|
.RS
|
|
Overflow customarily stops computation.
|
|
.br
|
|
Underflow is customarily flushed quietly to zero.
|
|
.br
|
|
CAUTION:
|
|
.RS
|
|
It is possible to have x
|
|
.if n \
|
|
!=
|
|
.if t \
|
|
\(!=
|
|
y and yet
|
|
x\-y = 0 because of underflow. Similarly
|
|
x > y > 0 cannot prevent either x\(**y = 0
|
|
or y/x = 0 from happening without warning.
|
|
.RE
|
|
.RE
|
|
Zero is represented ambiguously.
|
|
.RS
|
|
Although 2**55 different representations of zero are accepted by
|
|
the hardware, only the obvious representation is ever produced.
|
|
There is no \-0 on a VAX.
|
|
.RE
|
|
.If
|
|
is not part of the VAX architecture.
|
|
.br
|
|
Reserved operands:
|
|
.RS
|
|
of the 2**55 that the hardware
|
|
recognizes, only one of them is ever produced.
|
|
Any floating\-point operation upon a reserved
|
|
operand, even a MOVF or MOVD, customarily stops
|
|
computation, so they are not much used.
|
|
.RE
|
|
Exceptions:
|
|
.RS
|
|
Divisions by zero and operations that
|
|
overflow are invalid operations that customarily
|
|
stop computation or, in earlier machines, produce
|
|
reserved operands that will stop computation.
|
|
.RE
|
|
Rounding:
|
|
.RS
|
|
Every rational operation (+, \-, \(**, /) on a
|
|
VAX (but not necessarily on a PDP\-11), if not an
|
|
over/underflow nor division by zero, is rounded to
|
|
within half an \*(up, and when the rounding error is
|
|
exactly half an \*(up then rounding is away from 0.
|
|
.RE
|
|
.RE
|
|
.PP
|
|
Except for its narrow range, D_floating\-point is one of the
|
|
better computer arithmetics designed in the 1960's.
|
|
Its properties are reflected fairly faithfully in the elementary
|
|
functions for a VAX distributed in 4.3 BSD.
|
|
They over/underflow only if their results have to lie out of range
|
|
or very nearly so, and then they behave much as any rational
|
|
arithmetic operation that over/underflowed would behave.
|
|
Similarly, expressions like log(0) and atanh(1) behave
|
|
like 1/0; and sqrt(\-3) and acos(3) behave like 0/0;
|
|
they all produce reserved operands and/or stop computation!
|
|
The situation is described in more detail in manual pages.
|
|
.RS
|
|
.ll -0.5i
|
|
\fIThis response seems excessively punitive, so it is destined
|
|
to be replaced at some time in the foreseeable future by a
|
|
more flexible but still uniform scheme being developed to
|
|
handle all floating\-point arithmetic exceptions neatly.
|
|
See infnan(3M) for the present state of affairs.\fR
|
|
.ll +0.5i
|
|
.RE
|
|
.PP
|
|
How do the functions in 4.3 BSD's new \fIlibm\fR for UNIX
|
|
compare with their counterparts in DEC's VAX/VMS library?
|
|
Some of the VMS functions are a little faster, some are
|
|
a little more accurate, some are more puritanical about
|
|
exceptions (like pow(0.0,0.0) and atan2(0.0,0.0)),
|
|
and most occupy much more memory than their counterparts in
|
|
\fIlibm\fR.
|
|
The VMS codes interpolate in large table to achieve
|
|
speed and accuracy; the \fIlibm\fR codes use tricky formulas
|
|
compact enough that all of them may some day fit into a ROM.
|
|
.PP
|
|
More important, DEC regards the VMS codes as proprietary
|
|
and guards them zealously against unauthorized use. But the
|
|
\fIlibm\fR codes in 4.3 BSD are intended for the public domain;
|
|
they may be copied freely provided their provenance is always
|
|
acknowledged, and provided users assist the authors in their
|
|
researches by reporting experience with the codes.
|
|
Therefore no user of UNIX on a machine whose arithmetic resembles
|
|
VAX D_floating\-point need use anything worse than the new \fIlibm\fR.
|
|
.PP
|
|
\fBIEEE STANDARD 754 Floating\-Point Arithmetic:\fR
|
|
.PP
|
|
This standard is on its way to becoming more widely adopted
|
|
than any other design for computer arithmetic.
|
|
VLSI chips that conform to some version of that standard have been
|
|
produced by a host of manufacturers, among them ...
|
|
.nf
|
|
.ta 0.5i +\w'Intel i8070, i80287'u+6n
|
|
Intel i8087, i80287 National Semiconductor 32081
|
|
Motorola 68881 Weitek WTL-1032, ... , -1165
|
|
Zilog Z8070 Western Electric (AT&T) WE32106.
|
|
.ta
|
|
.fi
|
|
Other implementations range from software, done thoroughly
|
|
in the Apple Macintosh, through VLSI in the Hewlett\-Packard
|
|
9000 series, to the ELXSI 6400 running ECL at 3 Megaflops.
|
|
Several other companies have adopted the formats
|
|
of IEEE 754 without, alas, adhering to the standard's way
|
|
of handling rounding and exceptions like over/underflow.
|
|
The DEC VAX G_floating\-point format is very similar to the IEEE
|
|
754 Double format, so similar that the C programs for the
|
|
IEEE versions of most of the elementary functions listed
|
|
above could easily be converted to run on a MicroVAX, though
|
|
nobody has volunteered to do that yet.
|
|
.PP
|
|
The codes in 4.3 BSD's \fIlibm\fR for machines that conform to
|
|
IEEE 754 are intended primarily for the National Semi. 32081
|
|
and WTL 1164/65. To use these codes with the Intel or Zilog
|
|
chips, or with the Apple Macintosh or ELXSI 6400, is to
|
|
forego the use of better codes provided (perhaps freely) by
|
|
those companies and designed by some of the authors of the
|
|
codes above.
|
|
Except for \fIatan\fR, \fIcabs\fR, \fIcbrt\fR, \fIerf\fR,
|
|
\fIerfc\fR, \fIhypot\fR, \fIj0\-jn\fR, \fIlgamma\fR, \fIpow\fR
|
|
and \fIy0\-yn\fR,
|
|
the Motorola 68881 has all the functions in \fIlibm\fR on chip,
|
|
and faster and more accurate;
|
|
it, Apple, the i8087, Z8070 and WE32106 all use 64
|
|
.if n \
|
|
sig.
|
|
.if t \
|
|
significant
|
|
bits.
|
|
The main virtue of 4.3 BSD's
|
|
\fIlibm\fR codes is that they are intended for the public domain;
|
|
they may be copied freely provided their provenance is always
|
|
acknowledged, and provided users assist the authors in their
|
|
researches by reporting experience with the codes.
|
|
Therefore no user of UNIX on a machine that conforms to
|
|
IEEE 754 need use anything worse than the new \fIlibm\fR.
|
|
.PP
|
|
Properties of IEEE 754 Double\-Precision:
|
|
.RS
|
|
Wordsize: 64 bits, 8 bytes. Radix: Binary.
|
|
.br
|
|
Precision: 53
|
|
.if n \
|
|
sig.
|
|
.if t \
|
|
significant
|
|
bits, roughly like 16
|
|
.if n \
|
|
sig.
|
|
.if t \
|
|
significant
|
|
decimals.
|
|
.RS
|
|
If x and x' are consecutive positive Double\-Precision
|
|
numbers (they differ by 1 \*(up), then
|
|
.br
|
|
1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16.
|
|
.RE
|
|
.nf
|
|
.ta \w'Range:'u+1n +\w'Underflow threshold'u+1n +\w'= 2.0**1024'u+1n
|
|
Range: Overflow threshold = 2.0**1024 = 1.8e308
|
|
Underflow threshold = 0.5**1022 = 2.2e\-308
|
|
.ta
|
|
.fi
|
|
.RS
|
|
Overflow goes by default to a signed
|
|
.If "" .
|
|
.br
|
|
Underflow is \fIGradual,\fR rounding to the nearest
|
|
integer multiple of 0.5**1074 = 4.9e\-324.
|
|
.RE
|
|
Zero is represented ambiguously as +0 or \-0.
|
|
.RS
|
|
Its sign transforms correctly through multiplication or
|
|
division, and is preserved by addition of zeros
|
|
with like signs; but x\-x yields +0 for every
|
|
finite x. The only operations that reveal zero's
|
|
sign are division by zero and copysign(x,\(+-0).
|
|
In particular, comparison (x > y, x \(>= y, etc.)
|
|
cannot be affected by the sign of zero; but if
|
|
finite x = y then
|
|
.If
|
|
\&= 1/(x\-y)
|
|
.if n \
|
|
!=
|
|
.if t \
|
|
\(!=
|
|
\-1/(y\-x) =
|
|
.If \- .
|
|
.RE
|
|
.If
|
|
is signed.
|
|
.RS
|
|
it persists when added to itself
|
|
or to any finite number. Its sign transforms
|
|
correctly through multiplication and division, and
|
|
.If (finite)/\(+- \0=\0\(+-0
|
|
(nonzero)/0 =
|
|
.If \(+- .
|
|
But
|
|
.if n \
|
|
Infinity\-Infinity, Infinity\(**0 and Infinity/Infinity
|
|
.if t \
|
|
\(if\-\(if, \(if\(**0 and \(if/\(if
|
|
are, like 0/0 and sqrt(\-3),
|
|
invalid operations that produce \*(nn. ...
|
|
.RE
|
|
Reserved operands:
|
|
.RS
|
|
there are 2**53\-2 of them, all
|
|
called \*(nn (\fIN\fRot \fIa N\fRumber).
|
|
Some, called Signaling \*(nns, trap any floating\-point operation
|
|
performed upon them; they are used to mark missing
|
|
or uninitialized values, or nonexistent elements
|
|
of arrays. The rest are Quiet \*(nns; they are
|
|
the default results of Invalid Operations, and
|
|
propagate through subsequent arithmetic operations.
|
|
If x
|
|
.if n \
|
|
!=
|
|
.if t \
|
|
\(!=
|
|
x then x is \*(nn; every other predicate
|
|
(x > y, x = y, x < y, ...) is FALSE if \*(nn is involved.
|
|
.br
|
|
NOTE: Trichotomy is violated by \*(nn.
|
|
.RS
|
|
Besides being FALSE, predicates that entail ordered
|
|
comparison, rather than mere (in)equality,
|
|
signal Invalid Operation when \*(nn is involved.
|
|
.RE
|
|
.RE
|
|
Rounding:
|
|
.RS
|
|
Every algebraic operation (+, \-, \(**, /,
|
|
.if n \
|
|
sqrt)
|
|
.if t \
|
|
\(sr)
|
|
is rounded by default to within half an \*(up, and
|
|
when the rounding error is exactly half an \*(up then
|
|
the rounded value's least significant bit is zero.
|
|
This kind of rounding is usually the best kind,
|
|
sometimes provably so; for instance, for every
|
|
x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find
|
|
(x/3.0)\(**3.0 == x and (x/10.0)\(**10.0 == x and ...
|
|
despite that both the quotients and the products
|
|
have been rounded. Only rounding like IEEE 754
|
|
can do that. But no single kind of rounding can be
|
|
proved best for every circumstance, so IEEE 754
|
|
provides rounding towards zero or towards
|
|
.If +
|
|
or towards
|
|
.If \-
|
|
at the programmer's option. And the
|
|
same kinds of rounding are specified for
|
|
Binary\-Decimal Conversions, at least for magnitudes
|
|
between roughly 1.0e\-10 and 1.0e37.
|
|
.RE
|
|
Exceptions:
|
|
.RS
|
|
IEEE 754 recognizes five kinds of floating\-point exceptions,
|
|
listed below in declining order of probable importance.
|
|
.RS
|
|
.nf
|
|
.ta \w'Invalid Operation'u+6n +\w'Gradual Underflow'u+2n
|
|
Exception Default Result
|
|
.tc \(ru
|
|
|
|
.tc
|
|
Invalid Operation \*(nn, or FALSE
|
|
.if n \{\
|
|
Overflow \(+-Infinity
|
|
Divide by Zero \(+-Infinity \}
|
|
.if t \{\
|
|
Overflow \(+-\(if
|
|
Divide by Zero \(+-\(if \}
|
|
Underflow Gradual Underflow
|
|
Inexact Rounded value
|
|
.ta
|
|
.fi
|
|
.RE
|
|
NOTE: An Exception is not an Error unless handled
|
|
badly. What makes a class of exceptions exceptional
|
|
is that no single default response can be satisfactory
|
|
in every instance. On the other hand, if a default
|
|
response will serve most instances satisfactorily,
|
|
the unsatisfactory instances cannot justify aborting
|
|
computation every time the exception occurs.
|
|
.RE
|
|
.PP
|
|
For each kind of floating\-point exception, IEEE 754
|
|
provides a Flag that is raised each time its exception
|
|
is signaled, and stays raised until the program resets
|
|
it. Programs may also test, save and restore a flag.
|
|
Thus, IEEE 754 provides three ways by which programs
|
|
may cope with exceptions for which the default result
|
|
might be unsatisfactory:
|
|
.IP 1) \w'\0\0\0\0'u
|
|
Test for a condition that might cause an exception
|
|
later, and branch to avoid the exception.
|
|
.IP 2) \w'\0\0\0\0'u
|
|
Test a flag to see whether an exception has occurred
|
|
since the program last reset its flag.
|
|
.IP 3) \w'\0\0\0\0'u
|
|
Test a result to see whether it is a value that only
|
|
an exception could have produced.
|
|
.RS
|
|
CAUTION: The only reliable ways to discover
|
|
whether Underflow has occurred are to test whether
|
|
products or quotients lie closer to zero than the
|
|
underflow threshold, or to test the Underflow
|
|
flag. (Sums and differences cannot underflow in
|
|
IEEE 754; if x
|
|
.if n \
|
|
!=
|
|
.if t \
|
|
\(!=
|
|
y then x\-y is correct to
|
|
full precision and certainly nonzero regardless of
|
|
how tiny it may be.) Products and quotients that
|
|
underflow gradually can lose accuracy gradually
|
|
without vanishing, so comparing them with zero
|
|
(as one might on a VAX) will not reveal the loss.
|
|
Fortunately, if a gradually underflowed value is
|
|
destined to be added to something bigger than the
|
|
underflow threshold, as is almost always the case,
|
|
digits lost to gradual underflow will not be missed
|
|
because they would have been rounded off anyway.
|
|
So gradual underflows are usually \fIprovably\fR ignorable.
|
|
The same cannot be said of underflows flushed to 0.
|
|
.RE
|
|
.PP
|
|
At the option of an implementor conforming to IEEE 754,
|
|
other ways to cope with exceptions may be provided:
|
|
.IP 4) \w'\0\0\0\0'u
|
|
ABORT. This mechanism classifies an exception in
|
|
advance as an incident to be handled by means
|
|
traditionally associated with error\-handling
|
|
statements like "ON ERROR GO TO ...". Different
|
|
languages offer different forms of this statement,
|
|
but most share the following characteristics:
|
|
.IP \(em \w'\0\0\0\0'u
|
|
No means is provided to substitute a value for
|
|
the offending operation's result and resume
|
|
computation from what may be the middle of an
|
|
expression. An exceptional result is abandoned.
|
|
.IP \(em \w'\0\0\0\0'u
|
|
In a subprogram that lacks an error\-handling
|
|
statement, an exception causes the subprogram to
|
|
abort within whatever program called it, and so
|
|
on back up the chain of calling subprograms until
|
|
an error\-handling statement is encountered or the
|
|
whole task is aborted and memory is dumped.
|
|
.IP 5) \w'\0\0\0\0'u
|
|
STOP. This mechanism, requiring an interactive
|
|
debugging environment, is more for the programmer
|
|
than the program. It classifies an exception in
|
|
advance as a symptom of a programmer's error; the
|
|
exception suspends execution as near as it can to
|
|
the offending operation so that the programmer can
|
|
look around to see how it happened. Quite often
|
|
the first several exceptions turn out to be quite
|
|
unexceptionable, so the programmer ought ideally
|
|
to be able to resume execution after each one as if
|
|
execution had not been stopped.
|
|
.IP 6) \w'\0\0\0\0'u
|
|
\&... Other ways lie beyond the scope of this document.
|
|
.RE
|
|
.PP
|
|
The crucial problem for exception handling is the problem of
|
|
Scope, and the problem's solution is understood, but not
|
|
enough manpower was available to implement it fully in time
|
|
to be distributed in 4.3 BSD's \fIlibm\fR. Ideally, each
|
|
elementary function should act as if it were indivisible, or
|
|
atomic, in the sense that ...
|
|
.IP i) \w'iii)'u+2n
|
|
No exception should be signaled that is not deserved by
|
|
the data supplied to that function.
|
|
.IP ii) \w'iii)'u+2n
|
|
Any exception signaled should be identified with that
|
|
function rather than with one of its subroutines.
|
|
.IP iii) \w'iii)'u+2n
|
|
The internal behavior of an atomic function should not
|
|
be disrupted when a calling program changes from
|
|
one to another of the five or so ways of handling
|
|
exceptions listed above, although the definition
|
|
of the function may be correlated intentionally
|
|
with exception handling.
|
|
.PP
|
|
Ideally, every programmer should be able \fIconveniently\fR to
|
|
turn a debugged subprogram into one that appears atomic to
|
|
its users. But simulating all three characteristics of an
|
|
atomic function is still a tedious affair, entailing hosts
|
|
of tests and saves\-restores; work is under way to ameliorate
|
|
the inconvenience.
|
|
.PP
|
|
Meanwhile, the functions in \fIlibm\fR are only approximately
|
|
atomic. They signal no inappropriate exception except
|
|
possibly ...
|
|
.RS
|
|
Over/Underflow
|
|
.RS
|
|
when a result, if properly computed, might have lain barely within range, and
|
|
.RE
|
|
Inexact in \fIcabs\fR, \fIcbrt\fR, \fIhypot\fR, \fIlog10\fR and \fIpow\fR
|
|
.RS
|
|
when it happens to be exact, thanks to fortuitous cancellation of errors.
|
|
.RE
|
|
.RE
|
|
Otherwise, ...
|
|
.RS
|
|
Invalid Operation is signaled only when
|
|
.RS
|
|
any result but \*(nn would probably be misleading.
|
|
.RE
|
|
Overflow is signaled only when
|
|
.RS
|
|
the exact result would be finite but beyond the overflow threshold.
|
|
.RE
|
|
Divide\-by\-Zero is signaled only when
|
|
.RS
|
|
a function takes exactly infinite values at finite operands.
|
|
.RE
|
|
Underflow is signaled only when
|
|
.RS
|
|
the exact result would be nonzero but tinier than the underflow threshold.
|
|
.RE
|
|
Inexact is signaled only when
|
|
.RS
|
|
greater range or precision would be needed to represent the exact result.
|
|
.RE
|
|
.RE
|
|
.SH BUGS
|
|
When signals are appropriate, they are emitted by certain
|
|
operations within the codes, so a subroutine\-trace may be
|
|
needed to identify the function with its signal in case
|
|
method 5) above is in use. And the codes all take the
|
|
IEEE 754 defaults for granted; this means that a decision to
|
|
trap all divisions by zero could disrupt a code that would
|
|
otherwise get correct results despite division by zero.
|
|
.SH SEE ALSO
|
|
An explanation of IEEE 754 and its proposed extension p854
|
|
was published in the IEEE magazine MICRO in August 1984 under
|
|
the title "A Proposed Radix\- and Word\-length\-independent
|
|
Standard for Floating\-point Arithmetic" by W. J. Cody et al.
|
|
The manuals for Pascal, C and BASIC on the Apple Macintosh
|
|
document the features of IEEE 754 pretty well.
|
|
Articles in the IEEE magazine COMPUTER vol. 14 no. 3 (Mar.
|
|
1981), and in the ACM SIGNUM Newsletter Special Issue of
|
|
Oct. 1979, may be helpful although they pertain to
|
|
superseded drafts of the standard.
|